# NAME

Math::PlanePath::DigitGroups -- X,Y digits grouped by zeros

# SYNOPSIS

`````` use Math::PlanePath::DigitGroups;

my \$path = Math::PlanePath::DigitGroups->new (radix => 2);
my (\$x, \$y) = \$path->n_to_xy (123);``````

# DESCRIPTION

This path splits an N into X,Y by digit groups separated by a 0. The default is binary so for example

``    N = 110111001011``

is split into groups with a leading high 0 bit, and those groups then go to X and Y alternately,

``````    N = 11 0111 0 01 011
X   Y  X  Y  X

X = 11      0    011 = 110011
Y =    0111   01     =  11101``````

The result is a one-to-one mapping between numbers N>=0 and pairs X>=0,Y>=0.

The default binary is

``````    11  |   38   77   86  155  166  173  182  311  550  333  342  347
10  |   72  145  148  291  168  297  300  583  328  337  340  595
9  |   66  133  138  267  162  277  282  535  322  325  330  555
8  |  128  257  260  515  272  521  524 1031  320  545  548 1043
7  |   14   29   46   59  142   93  110  119  526  285  302  187
6  |   24   49   52   99   88  105  108  199  280  177  180  211
5  |   18   37   42   75   82   85   90  151  274  165  170  171
4  |   32   65   68  131   80  137  140  263  160  161  164  275
3  |    6   13   22   27   70   45   54   55  262  141  150   91
2  |    8   17   20   35   40   41   44   71  136   81   84   83
1  |    2    5   10   11   34   21   26   23  130   69   74   43
Y=0 |    0    1    4    3   16    9   12    7   64   33   36   19
+-------------------------------------------------------------
X=0    1    2    3    4    5    6    7    8    9   10   11``````

N=0,1,4,3,16,9,etc along the X axis is X with zero bits doubled. For example X=9 is binary 1001, double up the zero bits to 100001 for N=33 at X=9,Y=0. This is because in the digit groups Y=0 so when X is grouped by its zero bits there's an extra 0 from Y in between each group.

Similarly N=0,2,8,6,32,etc along the Y axis is Y with zero bits doubled, plus an extra zero bit at the low end coming from the first X=0 group. For example Y=9 is again binary 1001, doubled zeros to 100001, and an extra zero at the low end 1000010 is N=66 at X=0,Y=9.

The `radix => \$r` option selects a different base for the digit split. For example radix 5 gives

``````    radix => 5

12  |  60  301  302  303  304  685 1506 1507 1508 1509 1310 1511
11  |  55  276  277  278  279  680 1381 1382 1383 1384 1305 1386
10  | 250 1251 1252 1253 1254 1275 6256 6257 6258 6259 1300 6261
9  |  45  226  227  228  229  670 1131 1132 1133 1134 1295 1136
8  |  40  201  202  203  204  665 1006 1007 1008 1009 1290 1011
7  |  35  176  177  178  179  660  881  882  883  884 1285  886
6  |  30  151  152  153  154  655  756  757  758  759 1280  761
5  | 125  626  627  628  629  650 3131 3132 3133 3134  675 3136
4  |  20  101  102  103  104  145  506  507  508  509  270  511
3  |  15   76   77   78   79  140  381  382  383  384  265  386
2  |  10   51   52   53   54  135  256  257  258  259  260  261
1  |   5   26   27   28   29  130  131  132  133  134  255  136
Y=0 |   0    1    2    3    4   25    6    7    8    9   50   11
+-----------------------------------------------------------
X=0    1    2    3    4    5    6    7    8    9   10   11``````

## Real Line and Plane

This split is inspired by the digit grouping in the proof by Julius König that the real line is the same cardinality as the plane. (Cantor's original proof was a `ZOrderCurve` style digit interleaving.)

In König's proof a bijection between interval n=(0,1) and pairs x=(0,1),y=(0,1) is made by taking groups of digits stopping at a non-zero. Non-terminating fractions like 0.49999... are chosen over terminating 0.5000... so there's always infinitely many non-zero digits going downwards. For the integer form here the groupings are digit going upwards and there's infinitely many zero digits above the top, hence the grouping by zeros instead of non-zeros.

# FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

`\$path = Math::PlanePath::DigitGroups->new ()`
`\$path = Math::PlanePath::DigitGroups->new (radix => \$r)`

Create and return a new path object. The optional `radix` parameter gives the base for digit splitting (the default is binary, radix 2).

`(\$x,\$y) = \$path->n_to_xy (\$n)`

Return the X,Y coordinates of point number `\$n` on the path. Points begin at 0 and if `\$n < 0` then the return is an empty list.

# OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

``````    radix=2 (the default)
A084471    N on X axis, bit 0->00
A084472    N on X axis, in binary
A060142    N on X axis, sorted into ascending order``````

http://user42.tuxfamily.org/math-planepath/index.html